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This tag is for questions relating to 'Singular Value'. The term “singular value” relates to the distance between a matrix and the set of singular matrices

In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator $$~T : X → Y~$$ acting between Hilbert spaces $$~X~$$ and $$~Y~$$ , are the square roots of non-negative eigenvalues of the self-adjoint operator $$~T'~T~$$  (where $$~T'~$$ denotes the adjoint of $$~T~$$).

Definition: Let $$~A~$$ be an $$~m × n~$$ matrix and $$~λ_1,λ_2,~\cdots~, λ_n~$$ denote the eigenvalues of $$~A^{\text{T}}~ A~$$, with repetitions. Order these so that $$~λ_1 ≥ λ_2 ≥ \cdots ≥ λ_n ≥ 0~$$. Let $$~σ_i = \sqrt{λ_i~}~$$, so that $$~σ_1 ≥ σ_2 ≥\cdots ≥ σ_n ≥ 0~$$. The numbers $$~σ_1,~ σ_2 ,~\cdots ,~ σ_n~$$ are called singular values of $$~A~$$.

Singular values play an important role where the matrix is a transformation from one vector space to a different vector space, possibly with a different dimension.

• The number of nonzero singular values of $$~A~$$ equals the rank of $$~A~$$.
• In particular, if $$~A~$$ is an $$~m × n~$$ matrix with $$~m < n~$$ , then $$~A~$$ has at most $$~m~$$ nonzero singular values, because rank$$~(A) ≤ m~$$.
• Let $$~A~$$ be an $$~m × n~$$ matrix. Then the maximum value of $$~||Ax||~$$, where $$~x~$$ ranges over unit vectors in $$~\mathbb(R)^n~$$, is the largest singular value $$~σ_1~$$, and this is achieved when $$~x~$$ is an eigenvector of$$~A^{\text{T}}~ A~$$with eigenvalue $$~\sigma_1^2~$$. (This is the geometric significance of singular values.)

References:

https://en.wikipedia.org/wiki/Singular_value

http://mathworld.wolfram.com/SingularValue.html